clip-02-05.jl

Load Julia packages (libraries) needed for the snippets in chapter 0

using StatisticalRethinking, Optim
#gr(size=(600,600));

snippet 3.2

p_grid = range(0, step=0.001, stop=1)
prior = ones(length(p_grid))
likelihood = [pdf(Binomial(9, p), 6) for p in p_grid]
posterior = likelihood .* prior
posterior = posterior / sum(posterior)
samples = sample(p_grid, Weights(posterior), length(p_grid));
samples[1:5]

5-element Array{Float64,1}:
 0.727
 0.489
 0.607
 0.68 
 0.721

snippet 3.3

Draw 10000 samples from this posterior distribution

N = 10000
samples = sample(p_grid, Weights(posterior), N);

10000-element Array{Float64,1}:
 0.859
 0.567
 0.417
 0.701
 0.738
 0.562
 0.73 
 0.719
 0.618
 0.513
 ⋮    
 0.596
 0.594
 0.59 
 0.652
 0.684
 0.759
 0.436
 0.319
 0.551

In StatisticalRethinkingJulia samples will always be stored in an MCMCChains.Chains object.

chn = MCMCChains.Chains(reshape(samples, N, 1, 1), ["toss"]);

Object of type Chains, with data of type 10000×1×1 Array{Float64,3}

Iterations        = 1:10000
Thinning interval = 1
Chains            = 1
Samples per chain = 10000
parameters        = toss

2-element Array{ChainDataFrame,1}

Summary Statistics
. Omitted printing of 1 columns
│ Row │ parameters │ mean     │ std      │ naive_se   │ mcse       │ ess     │
│     │ Symbol     │ Float64  │ Float64  │ Float64    │ Float64    │ Any     │
├─────┼────────────┼──────────┼──────────┼────────────┼────────────┼─────────┤
│ 1   │ toss       │ 0.636631 │ 0.137942 │ 0.00137942 │ 0.00131646 │ 10405.3 │

Quantiles

│ Row │ parameters │ 2.5%     │ 25.0%   │ 50.0%   │ 75.0%   │ 97.5%   │
│     │ Symbol     │ Float64  │ Float64 │ Float64 │ Float64 │ Float64 │
├─────┼────────────┼──────────┼─────────┼─────────┼─────────┼─────────┤
│ 1   │ toss       │ 0.349975 │ 0.545   │ 0.645   │ 0.737   │ 0.878   │

Describe the chain

MCMCChains.describe(chn)

2-element Array{ChainDataFrame,1}

Summary Statistics
. Omitted printing of 1 columns
│ Row │ parameters │ mean     │ std      │ naive_se   │ mcse       │ ess     │
│     │ Symbol     │ Float64  │ Float64  │ Float64    │ Float64    │ Any     │
├─────┼────────────┼──────────┼──────────┼────────────┼────────────┼─────────┤
│ 1   │ toss       │ 0.636631 │ 0.137942 │ 0.00137942 │ 0.00131646 │ 10405.3 │

Quantiles

│ Row │ parameters │ 2.5%     │ 25.0%   │ 50.0%   │ 75.0%   │ 97.5%   │
│     │ Symbol     │ Float64  │ Float64 │ Float64 │ Float64 │ Float64 │
├─────┼────────────┼──────────┼─────────┼─────────┼─────────┼─────────┤
│ 1   │ toss       │ 0.349975 │ 0.545   │ 0.645   │ 0.737   │ 0.878   │

Plot the chain

plot(chn)

snippet 3.4

Create a vector to hold the plots so we can later combine them

p = Vector{Plots.Plot{Plots.GRBackend}}(undef, 2)
p[1] = scatter(1:N, samples, markersize = 2, ylim=(0.0, 1.3), lab="Draws")